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698 Chapter 11 Parametric Equations and Polar Coordinates

67. 68.

69. 70.

71. 72.

73. 74.

Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

Chapter 11 Practice Exercises 699

75. (a) Perihelion a ae a(1 e), Aphelion ea a a(1 e)œ � œ � œ � œ �

(b) Planet Perihelion Aphelion Mercury 0.3075 AU 0.4667 AUVenus 0.7184 AU 0.7282 AUEarth 0.9833 AU 1.0167 AUMars 1.3817 AU 1.6663 AUJupiter 4.9512 AU 5.4548 AUSaturn 9.0210 AU 10.0570 AUUranus 18.2977 AU 20.0623 AUNeptune 29.8135 AU 30.3065 AU

76. Mercury: r œ œ(0.3871) 1 0.2056

1 0.2056 cos 1 0.2056 cos 0.3707a b�

� �

#

) )

Venus: r œ œ(0.7233) 1 0.0068

1 0.0068 cos 1 0.0068 cos 0.7233a b�

� �

#

) )

Earth: r œ œ1 1 0.01671 0.0167 cos 1 0.0617 cos

0.9997a b�

� �

#

) )

Mars: r œ œ(1.524) 1 0.0934

1 0.0934 cos 1 0.0934 cos 1.511a b�

� �

#

) )

Jupiter: r œ œ(5.203) 1 0.0484

1 0.0484 cos 1 0.0484 cos 5.191a b�

� �

#

) )

Saturn: r œ œ(9.539) 1 0.0543

1 0.0543 cos 1 0.0543 cos 9.511a b�

� �

#

) )

Uranus: r œ œ(19.18) 1 0.0460

1 0.0460 cos 1 0.0460 cos 19.14a b�

� �

#

) )

Neptune: r œ œ(30.06) 1 0.0082

1 0.0082 cos 1 0.0082 cos 30.06a b�

� �

#

) )

CHAPTER 11 PRACTICE EXERCISES

1. x and y t 1 2x t y 2x 1 2. x t and y 1 t y 1 xœ œ � Ê œ Ê œ � œ œ � Ê œ �t#

È È

3. x tan t and y sec t x tan t 4. x 2 cos t and y 2 sin t x 4 cos t andœ œ Ê œ œ � œ Ê œ" " "

# #

# # # #

4

and y sec t 4x tan t and y 4 sin t x y 4# # # # # # # #"œ Ê œ œ Ê � œ4

4y sec t 4x 1 4y 4y 4x 1# # # # # #œ Ê � œ Ê � œ



5. x cos t and y cos t y ( x) x 6. x 4 cos t and y 9 sin t x 6 cos t andœ � œ Ê œ � œ œ œ Ê œ# # # # #

y 81 sin t 1# #œ Ê � œx16 81

y# #

7. 16x 9y 144 1 a 3 and b 4 x 3 cos t and y 4 sin t, 0 t 2# #� œ Ê � œ Ê œ œ Ê œ œ Ÿ Ÿx9 16

y# #

1

8. x y 4 x 2 cos t and y 2 sin t, 0 t 6# #� œ Ê œ � œ Ÿ Ÿ 1

9. x tan t, y sec t sin t sin ; tœ œ Ê œ œ œ œ Ê œ œ œ" "# # #

dy dy/dt dydx dx/dt sec t dx 3 3

sec t tan t sec t

tan t 3"

#

"

#

#¹

t 3œ Î1

1 1È

x tan and y sec 1 y x ; 2 cos t Ê œ œ œ œ Ê œ � œ œ œ Ê" " "# # # # œ Î

1 1

13 3 4 dx dx/dt dx

3 3 d y dy /dt d ycos t sec t

3

t 3

È È # w #

# #"

#

#¹

2 cosœ œ33 4

ˆ ‰1 "

10. x , y t (2) 3; t 2 x 1 andœ " � œ " � Ê œ œ œ � Ê œ � œ � œ Ê œ � œ" "

� # #t t dx dx/dt 2 dx 43 3 3 5dy dy/dt dy

# #

#

$

Š ‹Š ‹

3t

2t

¹t 2œ

y 1 y 3x ; t (2) 6œ � œ � Ê œ � � œ œ œ Ê œ œ3 3 3 34 dx dx/dt 4 dx 4

d y dy /dt d y

t 2# #" " �

�

$ $

œ

# w #

# ##

$

ˆ ‰Š ‹

3

2t

¹

11. (a) x 4t , y t 1 t y 1 1œ œ � Ê œ „ Ê œ „ � œ „ �2 3 x x2 2 8

3xÈ ÈŠ ‹ 3 2Î

(b) x cos t, y tan t sec t tan t 1 sec t y 1 yœ œ Ê œ Ê � œ Ê œ � œ Ê œ „1 1 1 xx x x x

2 2 2 1 x2 2

2 2� �È

12. (a) The line through 1, 2 with slope 3 is y 3x 5 x t, y 3t 5, ta b� œ � Ê œ œ � �_ � � _

(b) x 1 y 2 9 x 1 3 cos t, y 2 3 sin t x 1 3 cos t, y 2 3 sin t, 0 t 2a b a b� � � œ Ê � œ � œ Ê œ � œ � � Ÿ Ÿ2 21

(c) y 4x x x t, y 4t t, tœ � Ê œ œ � �_ � � _2 2

(d) 9x 4y 36 1 x 2 cos t, y 3 sin t, 0 t 22 2 x4 9

y� œ Ê � œ Ê œ œ Ÿ Ÿ2 2

1

13. y x x x 2 x L 1 2 x dxœ � Ê œ � Ê œ � � Ê œ � � �"Î# �"Î# "Î#" " " " " "# #

#x

3 dx dx 4 x 4 xdy dy$Î# Š ‹ ˆ ‰ ˆ ‰É'

1

4

L 2 x dx x x dx x x dx 2x xÊ œ � � œ � œ � œ �' ' '1 1 1

4 4 4É ˆ ‰ ˆ ‰ � ‘É a b" " " " "�"Î# "Î# #

# #�"Î# "Î# "Î# $Î# %

"4 x 4 32

4 8 2 2œ � � � œ � œ" "# #� ‘ ˆ ‰ˆ ‰ ˆ ‰2 2 14 10

3 3 3 3†

14. x y x L 1 dy 1 dyœ Ê œ Ê œ Ê œ � œ �#Î$ �"Î$# #

dx 2 dx 4x dx 4dy 3 dy 9 dy 9xŠ ‹ Š ‹Ê É�#Î$

#Î$' '

1 1

8 8

dx 9x 4 x dx; u 9x 4 du 6y dy; x 1 u 13,œ œ � œ � Ê œ œ Ê œ' '1 1

8 8È9x 43x 3

#Î$

"Î$� " #Î$ �"Î$ #Î$ �"Î$È ˆ ‰ �

x 8 u 40 L u du u 40 13 7.634d � ‘ � ‘œ Ê œ Ä œ œ œ � ¸" " ""Î# $Î# $Î# $Î#%!

"$ #18 18 3 72'

13

40

15. y x x x x x 2 xœ � Ê œ � Ê œ � �5 512 8 dx dx 4

dy dy'Î& %Î& "Î& �"Î& #Î& �#Î&" " "# #

#Š ‹ ˆ ‰ L 1 x 2 x dx L x 2 x dx x x dxÊ œ � � � Ê œ � � œ �' '

1 1

32 32 32

1

É Éa b a b a b' É" " "#Î& �#Î& #Î& �#Î& "Î& �"Î& #

4 4 4



x x dx x x 2 2œ � œ � œ � � � œ �'1

32" " " "# # # #

"Î& �"Î& 'Î& %Î& ' %$#

"ˆ ‰ � ‘ � ‘ ˆ ‰ˆ ‰ ˆ ‰5 5 5 5 5 5 315 75

6 4 6 4 6 4 6 4† †

(1260 450)œ � œ œ"48 48 8

1710 285

16. x y y y L 1 y dyœ � Ê œ � Ê œ � � Ê œ � � �" " " " " " " " " "# # #

$ # %#

%1 y dy 4 y dy 16 16

dx dxy y# % %Š ‹ Š ‹Ê'

1

2

y dy y dy y dy yœ � � œ � œ � œ �' ' '1 1 1

2 2 2É ÊŠ ‹ Š ‹ ’ “" " " " " " " " "% ## #

# ## $

"16 4 y 4 y 1 yy% # #

1œ � � � œ � œˆ ‰ ˆ ‰8 7 1312 1 1 12

" " "# # # #

17. 5 sin t 5 sin 5t and 5 cos t 5 cos 5tdx dxdt dt dt dt

dy dyœ � � œ � Ê �Êˆ ‰ Š ‹# #

5 sin t 5 sin 5t 5 cos t 5 cos 5tœ � � � �Éa b a b# #

5 sin 5t sin t sin 5t sin t cos t cos t cos 5t cos 5t sin t sin 5t cos t cos 5 tœ � # � � � # � œ & # � # �È È a b# # # #

5 cos t 5 cos t sin t sin t sin t (since t )œ # " � % œ % " � % œ "! # œ "!l # l œ "! # ! Ÿ ŸÈ a b a bÉ ˆ ‰ È"# #

# 1

Length sin t dt 5 cos tÊ œ "! # œ � # œ �& �" � �& " œ "!'!

Î Î#!

11

2 c d a ba b a ba b

18. 3t 12t and 3t 12t 3t 12t 3t 12t 288t 8tdx dxdt dt dt dt

2 2dy dy 2 2 4œ � œ � Ê � œ � � � œ � "Êˆ ‰ Š ‹ Éa b a b È# ## # #

3 2 t 16 t Length 3 2 t 16 t dt 3 2 t 16 t dt; u 16 t du 2t dtœ � Ê œ � œ � œ � Ê œÈ È Èk k k kÈ È È ’2 2 2 2' '! !

" "

du t dt; t 0 u 16; t 1 u 17 ; u du u 17 16Ê œ œ Ê œ œ Ê œ œ œ �"#

"“ È � ‘ Š ‹a b a b3 2 3 2 3 22 2 3 2 3 316

72 2 23/2 17

163/2 3/2È È È'

17 64 2 17 64 8.617.œ † � œ � ¸3 22 3

2 3/2 3/2È Š ‹ Š ‹a b a bÈ

19. sin and cos sin cos sin cos dx dxd d d d

dy dy) ) ) )œ �$ œ $ Ê � œ �$ � $ œ $ � œ $) ) ) ) ) )Êˆ ‰ Š ‹ Éa b a b a bÈ# #

# # # #

Length d dÊ œ $ œ $ œ $ � ! œ' '! !

$ Î $ Î$ *# #

1 11 1

2 2

) ) ˆ ‰

20. x t and y t, 3 t 3 2t and t Length 2t t dtœ œ � � Ÿ Ÿ Ê œ œ � " Ê œ � � "# #

�

# ##t dx3 dt dt

dy

3

3$ È È Éa b a b'

È

È

t t dt t 2t dt t dt t dt tœ � # � " œ � � " œ � " œ � " œ �' ' '� � �

% # % # # #

�

#

�È È È

È È È

ÈÈ È

È3 3 3

3 3 3

3

3t3

3

3È È Éa b a b ’ “' 3

4 3œ È

21. x and y 2t, 0 t 5 t and 2 Surface Area 2 (2t) t 4 dt 2 u duœ œ Ÿ Ÿ Ê œ œ Ê œ � œt dxdt dt

dy0 4

5 9#

## "Î#È È' '

È1 1

2 u , where u t 4 du 2t dt; t 0 u 4, t 5 u 9œ œ œ � Ê œ œ Ê œ œ Ê œ1 � ‘ È2 763 3

$Î# #*

%1

22. x t and y 4 t , t 1 2t and œ � œ Ÿ Ÿ Ê œ � œ# " " "2t dt 2t dt2

dx 2dyt

È È È#

Surface Area 2 t 2t dt 2 t 2t dtÊ œ � � � œ � �' '1 2 1 2

1 1

t 2t t t2

tÎ Î# #" " " "

# # #

# ##

È ÈÈ1 1ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰Ê Š ‹ É# #

2 t 2t dt 2 2t t dt 2 t t tœ � � œ � � œ � �1 1 1' '1 2 1 2

1 1

2t 2t 4 2 83 3

Î Î# $ �$ % �#" " " " "

# #

"

"Î #È È Èˆ ‰ ˆ ‰ ˆ ‰ � ‘#

2 2œ �1 Š ‹3 24

È



23. r cos 2 3 r cos cos sin sin ˆ ‰ ˆ ‰È) ) )� œ Ê �1 1 13 3 3

2 3 r cos r sin 2 3œ Ê � œÈ È"# #) )

È3

r cos 3 r sin 4 3 x 3 y 4 3Ê � œ Ê � œ) )È È È È y x 4Ê œ �

È33

24. r cos r cos cos sin sin ˆ ‰ ˆ ‰) ) )� œ Ê �3 3 34 4 4

21 1 1È#

r cos r sin x y 1œ Ê � � œ Ê � � œÈ È È È2 2 2 2# # # #) )

y x 1Ê œ �

25. r 2 sec r r cos 2 x 2œ Ê œ Ê œ Ê œ) )2cos )

26. r 2 sec r cos 2 x 2œ � Ê œ � Ê œ �È È È) )

27. r csc r sin yœ � Ê œ � Ê œ �3 3 3# # #) )



28. r 3 3 csc r sin 3 3 y 3 3œ Ê œ Ê œÈ È È) )

29. r 4 sin r 4r sin x y 4y 0œ � Ê œ � Ê � � œ) )# # #

x (y 2) 4; circle with center ( 2) andÊ � � œ !ß�# #

radius 2.

30. r 3 3 sin r 3 3 r sin œ Ê œÈ È) )#

x y 3 3 y 0 x y ;Ê � � œ Ê � � œ# # ##

#È Š ‹3 3 274

È

circle with center and radius Š ‹!ß 3 3 3 3È È# #

31. r 2 2 cos r 2 2 r cos œ Ê œÈ È) )#

x y 2 2 x 0 x 2 y 2;Ê � � œ Ê � � œ# # ##È ÈŠ ‹

circle with center 2 0 and radius 2Š ‹È Èß

32. r 6 cos r 6r cos x y 6x 0œ � Ê œ � Ê � � œ) )# # #

(x 3) y 9; circle with center ( 3 0) andÊ � � œ � ß# #

radius 3



33. x y 5y 0 x y C# # ## #

#� � œ Ê � � œ Ê œ !ß�ˆ ‰ ˆ ‰5 25 5

4

and a ; r 5r sin 0 r 5 sin œ � œ Ê œ �5#

# ) )

34. x y 2y 0 x (y 1) 1 C ( 1) and# # # #� � œ Ê � � œ Ê œ !ß

a 1; r 2r sin 0 r 2 sin œ � œ Ê œ# ) )

35. x y 3x 0 x y C# # ## #

#� � œ Ê � � œ Ê œ ß !ˆ ‰ ˆ ‰3 9 3

4

and a ; r 3r cos 0 r 3 cos œ � œ Ê œ3#

# ) )

36. x y 4x 0 (x 2) y 4 C ( 2 0)# # # #� � œ Ê � � œ Ê œ � ß

and a 2; r 4r cos 0 r 4 cos œ � œ Ê œ �# ) )

37. 38.

39. d 40. e 41. l 42. f

43. k 44. h 45. i 46. j



47. A 2 r d (2 cos ) d 4 4 cos cos d 4 4 cos dœ œ � œ � � œ � �' ' ' '0 0 0 0

1 1 1 1

" �# #

# # #) ) ) ) ) ) ) )a b ˆ ‰1 cos 2)

4 cos d 4 sin œ � � œ � � œ'0

1ˆ ‰ � ‘9 cos 2 9 sin 2 92 4# # #!

) ) ) ) 1) ) 1

48. A sin 3 d d sin 6œ œ œ � œ' '0 0

3 31 1Î Î" � " "# #

# Î$

!a b ˆ ‰ � ‘) ) ) ) )1 cos 6

4 6 12) 11

49. r 1 cos 2 and r 1 1 1 cos 2 0 cos 2 2 ; thereforeœ � œ Ê œ � Ê œ Ê œ Ê œ) ) ) ) )1 1# 4

A 4 (1 cos 2 ) 1 d 2 1 2 cos 2 cos 2 1 dœ � � œ � � �' '0 0

4 41 1Î Î"#

# # #c d a b) ) ) ) )

2 2 cos 2 d 2 sin 2 2 1 0 2œ � � œ � � œ � � œ �'0

41Î ˆ ‰ � ‘ ˆ ‰) ) ) )" "# #

Î%

!cos 4 sin 4

2 8 8 4) ) 1 11

50. The circle lies interior to the cardioid. Thus,

A 2 [2(1 sin )] d (the integral is the area of the cardioid minus the area of the circle)œ � �'� Î

Î

1

1

2

2"#

#) ) 1

4 1 2 sin sin d (6 8 sin 2 cos 2 ) d 6 8 cos sin 2œ � � � œ � � � œ � � �' '� Î � Î

Î Î

1 1

1 1

2 2

2 2a b c d) ) ) 1 ) ) ) 1 ) ) ) 1# Î#� Î#1

1

3 ( 3 ) 5œ � � � œc d1 1 1 1

51. r 1 cos sin ; Length ( 1 cos ) ( sin ) d 2 2 cos dœ � � Ê œ � œ � � � � œ �) ) ) ) ) ) )drd)

' '0 0

2 21 1È È# #

d 2 sin d 4 cos ( 4)( 1) ( 4)(1) 8œ œ œ � œ � � � � œ' '0 0

2 21 1É � ‘4(1 cos )2

�# #

#

!) ) ) 1

) )

52. r 2 sin 2 cos , 0 2 cos 2 sin ; r (2 sin 2 cos ) (2 cos 2 sin )œ � Ÿ Ÿ Ê œ � � œ � � �) ) ) ) ) ) ) ) )1) )#

# # ##dr drd d

ˆ ‰ 8 sin cos 8 L 8 d 2 2 2 2 2œ � œ Ê œ œ œ œa b È ’ “È È Èˆ ‰# #

Î#

! #) ) ) ) 1'0

21Î 11

53. r 8 sin , 0 8 sin cos ; r 8 sin 8 sin cos œ Ÿ Ÿ Ê œ � œ �$ # # $ ## # #ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ � ‘ � ‘ˆ ‰ ˆ ‰ ˆ ‰) 1 ) ) ) ) )) )3 4 d 3 3 d 3 3 3

dr dr)

64 sin L 64 sin d 8 sin d 8 dœ Ê œ œ œ% #% �

#ˆ ‰ ˆ ‰ ˆ ‰É ’ “) ) )

3 3 31 cos' ' '

0 0 0

4 4 41 1 1Î Î Î

) ) )ˆ ‰2

3)

4 4 cos d 4 6 sin 4 6 sin 0 3œ � œ � œ � � œ �'0

41Î � ‘ � ‘ ˆ ‰ ˆ ‰ˆ ‰ ˆ ‰2 23 3 4 6) ) 1 11

) ) 1Î%

!

54. r 1 cos 2 (1 cos 2 ) ( 2 sin 2 ) œ � Ê œ � � œ Ê œÈ ˆ ‰) ) )dr sin 2 dr sin 2d d 1 cos 21 cos 2) ) )

) )

)

" �# �

�"Î#�

#

È#

r 1 cos 2Ê � œ � � œ œ# #

� � �� ˆ ‰dr sin 2 1 2 cos 2 cos 2 sin 2

d 1 cos 2 1 cos 2 1 cos 2(1 cos 2 ) sin 2

) ) ) )) ) ) )) )

)# # ## #

2 L 2 d 2 2œ œ Ê œ œ � � œ2 2 cos 21 cos 2�� # #

) 1 1)

'� Î

Î

1

1

2

2 È È È� ‘ˆ ‰) 1

55. x 4y y 4p 4 p 1; 56. x 2y y 4p 2 p ;# ## #

"œ � Ê œ � Ê œ Ê œ œ Ê œ Ê œ Ê œx x4

# #

therefore Focus is (0 1), Directrix is y 1 therefore Focus is ; Directrix is yß � œ !ß œ �ˆ ‰" "# #



57. y 3x x 4p 3 p ; 58. y x x 4p p ;# #œ Ê œ Ê œ Ê œ œ � Ê œ � Ê œ Ê œy y3 4 3 3 3

3 8 8 2# #

ˆ ‰83

therefore Focus is 0 , Directrix is x therefore Focus is , Directrix is xˆ ‰ ˆ ‰3 3 2 24 4 3 3ß œ � � ß ! œ

59. 16x 7y 112 1 60. x 2y 4 1 c 4 2 2# # # # ##� œ Ê � œ � œ Ê � œ Ê œ � œx x

7 16 4y y# ## #

c 16 7 9 c 3; e c 2 ; eÊ œ � œ Ê œ œ œ Ê œ œ œ##

c 3 ca 4 a

2È È

61. 3x y 3 x 1 c 1 3 4 62. 5y 4x 20 1 c 4 5 9# # # # # # #� œ Ê � œ Ê œ � œ � œ Ê � œ Ê œ � œy y3 4 5

x# # #

c 2; e 2; the asymptotes are c 3, e ; the asymptotes are y xÊ œ œ œ œ Ê œ œ œ œ „c 2 c 3 2a 1 a 5# È

y 3 xœ „È

63. x 12y y 4p 12 p 3 focus is ( 3), directrix is y 3, vertex is (0 0); therefore new##œ � Ê � œ Ê œ Ê œ Ê !ß� œ ßx

1

#

vertex is (2 3), new focus is (2 0), new directrix is y 6, and the new equation is (x 2) 12(y 3)ß ß œ � œ � �#

64. y 10x x 4p 10 p focus is 0 , directrix is x , vertex is (0 0); therefore new## # #œ Ê œ Ê œ Ê œ Ê ß œ � ßy

105 5 5# ˆ ‰

vertex is 1 , new focus is (2 1), new directrix is x 3, and the new equation is (y 1) 10 xˆ ‰ ˆ ‰� ß� ß� œ � � œ �" "# #

#

65. 1 a 5 and b 3 c 25 9 4 foci are 4 , vertices are 5 , center isx9 5

y# #

� œ Ê œ œ Ê œ � œ Ê !ß „ !ß „#È a b a b

(0 0); therefore the new center is ( 5), new foci are ( 3 1) and ( 3 9), new vertices are ( 10) andß �$ß� � ß� � ß� �$ß�

( 0), and the new equation is 1�$ß � œ(x 3) (y 5)9 5� �

#

# #



66. 1 a 13 and b 12 c 169 144 5 foci are 5 0 , vertices are 13 0 , centerx169 144

y# #

� œ Ê œ œ Ê œ � œ Ê „ ß „ ßÈ a b a b is (0 0); therefore the new center is (5 12), new foci are (10 12) and (0 12), new vertices are (18 12) andß ß ß ß ß

( 8 12), and the new equation is 1� ß � œ(x 5) (y 12)169 144� �# #

67. 1 a 2 2 and b 2 c 8 2 10 foci are 0 10 , vertices arey8 2

x# #

� œ Ê œ œ Ê œ � œ Ê ß „È È È È ÈŠ ‹ 0 2 2 , center is (0 0), and the asymptotes are y 2x; therefore the new center is 2 2 2 , new foci areŠ ‹ Š ‹È Èß „ ß œ „ ß

2 2 2 10 , new vertices are 2 4 2 and ( 0), the new asymptotes are y 2x 4 2 2 andŠ ‹ Š ‹È È ÈÈß „ ß #ß œ � �

y 2x 4 2 2; the new equation is 1œ � � � � œÈ Š ‹Èy 2 2

8(x 2)� �

#

#

#

68. 1 a 6 and b 8 c 36 64 10 foci are 10 0 , vertices are 6 0 , the centerx36 64

y# #

� œ Ê œ œ Ê œ � œ Ê „ ß „ ßÈ a b a b is (0 0) and the asymptotes are or y x; therefore the new center is ( 10 3), the new foci areß œ „ œ „ � ß�y

8 6 3x 4

( 20 3) and (0 3), the new vertices are ( 16 3) and ( 4 3), the new asymptotes are y x and� ß� ß� � ß� � ß� œ �4 313 3

y x ; the new equation is 1œ � � � œ4 493 3 36 64

(x 10) (y 3)� �# #

69. x 4x 4y 0 x 4x 4 4y 4 (x 2) 4y 4 y 1, a hyperbola; a 2 and# # # # # # #�� œ Ê � � � œ Ê � � œ Ê � œ œ(x 2)4

#

b 1 c 1 4 5 ; the center is (2 0), the vertices are ( 0) and (4 0); the foci are 2 5 0 andœ Ê œ � œ ß !ß ß „ ßÈ È ÈŠ ‹ the asymptotes are y œ „ x 2�

#

70. 4x y 4y 8 4x y 4y 4 4 4x (y 2) 4 x 1, a hyperbola; a 1 and# # # # # # # �� œ Ê � � � œ Ê � � œ Ê � œ œ(y 2)4

#

b 2 c 1 4 5 ; the center is ( 2), the vertices are (1 2) and ( 2), the foci are 5 2 andœ Ê œ � œ !ß ß �"ß „ ßÈ È ÈŠ ‹ the asymptotes are y 2x 2œ „ �

71. y 2y 16x 49 y 2y 1 16x 48 (y 1) 16(x 3), a parabola; the vertex is ( 1);# # #� � œ � Ê � � œ � � Ê � œ � � �$ß

4p 16 p 4 the focus is ( 7 1) and the directrix is x 1œ Ê œ Ê � ß œ

72. x 2x 8y 17 x 2x 1 8y 16 (x 1) 8(y 2), a parabola; the vertex is (1 2);# # #� � œ � Ê � � œ � � Ê � œ � � ß�

4p 8 p 2 the focus is (1 4) and the directrix is y 0œ Ê œ Ê ß� œ

73. 9x 16y 54x 64y 1 9 x 6x 16 y 4y 1 9 x 6x 9 16 y 4y 4 144# # # # # #� � � œ � Ê � � � œ � Ê � � � � � œa b a b a b a b 9(x 3) 16(y 2) 144 1, an ellipse; the center is ( 3 2); a 4 and b 3Ê � � � œ Ê � œ � ß œ œ# # � �(x 3) (y 2)

16 9

# #

c 16 9 7 ; the foci are 7 2 ; the vertices are (1 2) and ( 7 2)Ê œ � œ �$ „ ß ß � ßÈ È ÈŠ ‹

74. 25x 9y 100x 54y 44 25 x 4x 9 y 6y 44 25 x 4x 4 9 y 6y 9 225# # # # # #� � � œ Ê � � � œ Ê � � � � � œa b a b a b a b 1, an ellipse; the center is (2 3); a 5 and b 3 c 25 9 4; the foci areÊ � œ ß� œ œ Ê œ � œ(x 2) (y 3)

9 25� �# # È

(2 1) and (2 7); the vertices are (2 2) and (2 8)ß ß � ß ß�

75. x y 2x 2y 0 x 2x 1 y 2y 1 2 (x 1) (y 1) 2, a circle with center (1 1) and# # # # # #� � � œ Ê � � � � � œ Ê � � � œ ß

radius 2œ È

76. x y 4x 2y 1 x 4x 4 y 2y 1 6 (x 2) (y 1) 6, a circle with center ( 2 1)# # # # # #� � � œ Ê � � � � � œ Ê � � � œ � ß�

and radius 6œ È



77. r e 1 parabola with vertex at (1 0)œ Ê œ Ê ß21 cos � )

78. r r e ellipse;œ Ê œ Ê œ Ê8 42 cos 1 cos � #�

") )ˆ ‰"

#

ke 4 k 4 k 8; k ea 8 aœ Ê œ Ê œ œ � Ê œ �" "# #

a ae ˆ ‰"

#

a ea ; therefore the center isÊ œ Ê œ œ16 16 83 3 3

ˆ ‰ ˆ ‰"#

; vertices are ( ) and 0ˆ ‰ ˆ ‰8 83 3ß )ß ß1 1

79. r e 2 hyperbola; ke 6 2k 6œ Ê œ Ê œ Ê œ61 2 cos � )

k 3 vertices are (2 ) and (6 )Ê œ Ê ß ß1 1

80. r r e ; ke 4œ Ê œ Ê œ œ12 43 sin 31 sin � �

") )ˆ ‰"

3

k 4 k 12; a 1 e 4 a 1Ê œ Ê œ � œ Ê �" "# #

3 3a b ’ “ˆ ‰ 4 a ea ; therefore theœ Ê œ Ê œ œ9 9 3

3# # #"ˆ ‰ ˆ ‰

center is ; vertices are 3 and 6ˆ ‰ ˆ ‰ ˆ ‰3 3 3# # # #ß ß ß1 1 1

81. e 2 and r cos 2 x 2 is directrix k 2; the conic is a hyperbola; r rœ œ Ê œ Ê œ œ Ê œ) ke1 e cos 1 cos

(2)(2)� �#) )

rÊ œ 41 cos �# )

82. e 1 and r cos 4 x 4 is directrix k 4; the conic is a parabola; r rœ œ � Ê œ � Ê œ œ Ê œ) ke1 e cos 1 cos

(4)(1)� �) )

rÊ œ 41 cos � )

83. e and r sin 2 y 2 is directrix k 2; the conic is an ellipse; r rœ œ Ê œ Ê œ œ Ê œ"# � �

) ke1 e sin

(2)1 sin ) )

ˆ ‰ˆ ‰

"

#

"

#

rÊ œ 22 sin � )

84. e and r sin 6 y 6 is directrix k 6; the conic is an ellipse; r rœ œ � Ê œ � Ê œ œ Ê œ"� �3 1 e sin

ke (6)1 sin

)) )

ˆ ‰ˆ ‰

"

"

3

3

rÊ œ 63 sin � )

85. (a) Around the x-axis: 9x 4y 36 y 9 x y 9 x and we use the positive root:# # # # #� œ Ê œ � Ê œ „ �9 94 4É

V 2 9 x dx 2 9 x dx 2 9x x 24œ � œ � œ � œ' '0 0

2 2

1 1 1 1Š ‹É ˆ ‰ � ‘9 9 34 4 4

##

# $ #

!


Chapter 11 Additional and Advanced Exercises 709

(b) Around the y-axis: 9x 4y 36 x 4 y x 4 y and we use the positive root:# # # # #� œ Ê œ � Ê œ „ �4 49 9É

V 2 4 y dy 2 4 y dy 2 4y y 16œ � œ � œ � œ' '0 0

3 3

1 1 1 1Š ‹É ˆ ‰ � ‘4 4 49 9 27

##

# $ $

!

86. 9x 4y 36, x 4 y y x 4 ; V x 4 dx x 4 dx# # # #�# #

# ##

� œ œ Ê œ Ê œ � œ � œ �9x 36 3 3 94 4

# È ÈŠ ‹ a b' '2 2

4 4

1 1

4x 16 8 (32) 24œ � œ � � � œ � œ œ9 x 9 64 8 9 56 24 34 3 4 3 3 4 3 3 41 1 1 1’ “ � ‘ ˆ ‰ˆ ‰ ˆ ‰$

%

#1

87. (a) r r er cos k x y ex k x y k ex x yœ Ê � œ Ê � � œ Ê � œ � Ê �k1 e cos �

# # # # # #)

) È È k 2kex e x x e x y 2kex k 0 1 e x y 2kex k 0œ � � Ê � � � � œ Ê � � � � œ# # # # # # # # # # # #a b (b) e 0 x y k 0 x y k circle;œ Ê � � œ Ê � œ Ê# # # # # #

0 e 1 e 1 e 1 0 B 4AC 0 4 1 e (1) 4 e 1 0 ellipse;� � Ê � Ê � � Ê � œ � � œ � � Ê# # # # # #a b a b e 1 B 4AC 0 4(0)(1) 0 parabola;œ Ê � œ � œ Ê# #

e 1 e 1 B 4AC 0 4 1 e (1) 4e 4 0 hyperbola� Ê � Ê � œ � � œ � � Ê# # # # #a b88. Let (r ) be a point on the graph where r a . Let (r ) be on the graph where r a and" " " " # # # #ß œ ß œ) ) ) )

2 . Then r and r lie on the same ray on consecutive turns of the spiral and the distance between) ) 1# " " #œ �

the two points is r r a a a( ) 2 a, which is constant.# " # " # "� œ � œ � œ) ) ) ) 1

CHAPTER 11 ADDITIONAL AND ADVANCED EXERCISES

1. Directrix x 3 and focus (4 0) vertex is œ ß Ê ß !ˆ ‰7#

p the equation is xÊ œ Ê � œ"# # #

7 y#

2. x 6x 12y 9 0 x 6x 9 12y y vertex is (3 0) and p 3 focus is (3 3) and the# # �� œ Ê � � œ Ê œ Ê ß œ Ê ß(x 3)12

#

directrix is y 3œ �

3. x 4y vertex is ( 0) and p 1 focus is ( 1); thus the distance from P(x y) to the vertex is x y# # #œ Ê !ß œ Ê !ß ß �È and the distance from P to the focus is x (y 1) x y 2 x (y 1)È È È# # # # # #� � Ê � œ � �

x y 4 x (y 1) x y 4x 4y 8y 4 3x 3y 8y 4 0, which is a circleÊ � œ � � Ê � œ � � � Ê � � � œ# # # # # # # # # #c d 4. Let the segment a b intersect the y-axis in point A and�

intersect the x-axis in point B so that PB b and PA aœ œ

(see figure). Draw the horizontal line through P and let it intersect the y-axis in point C. Let PBOn œ )

APC . Then sin and cos Ê n œ œ œ) ) )yb a

x

cos sin 1.Ê � œ � œxa b

y#

# #

## #) )

5. Vertices are 2 a 2; e 0.5 c 1 foci are 0 1a b a b!ß „ Ê œ œ Ê œ Ê œ Ê ß „c ca #



6. Let the center of the ellipse be (x 0); directrix x 2, focus (4 0), and e c 2 2 cß œ ß œ Ê � œ Ê œ �2 a a3 e e

a (2 c). Also c ae a a 2 a a a a a ; x 2Ê œ � œ œ Ê œ � Ê œ � Ê œ Ê œ � œ2 2 2 2 4 4 5 4 12 a3 3 3 3 3 9 9 3 5 e

ˆ ‰ x 2 x the center is 0 ; x 4 c c 4 so that c a bÊ � œ œ Ê œ Ê ß � œ Ê œ � œ œ �ˆ ‰ ˆ ‰ ˆ ‰12 3 18 28 28 28 8

5 5 5 5 5 5## # #

; therefore the equation is 1 or 1œ � œ � œ � œˆ ‰ ˆ ‰12 8 805 5 25 144 16

x 25 xy 5y# # � �ˆ ‰ ˆ ‰ˆ ‰ ˆ ‰

28 285 5

144 8025 25

# ## #

7. Let the center of the hyperbola be (0 y).ß

(a) Directrix y 1, focus (0 7) and e 2 c 6 c 6 a 2c 12. Also c ae 2aœ � ß� œ Ê � œ Ê œ � Ê œ � œ œa ae e

a 2(2a) 12 a 4 c 8; y ( 1) 2 y 1 the center is (0 1); c a bÊ œ � Ê œ Ê œ � � œ œ œ Ê œ Ê ß œ �a 4e #

# # #

b c a 64 16 48; therefore the equation is 1Ê œ � œ � œ � œ# # # �(y 1)16 48

x# #

(b) e 5 c 6 c 6 a 5c 30. Also, c ae 5a a 5(5a) 30 24a 30 aœ Ê � œ Ê œ � Ê œ � œ œ Ê œ � Ê œ Ê œa a 5e e 4

c ; y ( 1) y the center is ; c a b b c aÊ œ � � œ œ œ Ê œ � Ê !ß� œ � Ê œ �25 a 3 34 e 5 4 4 4

ˆ ‰54 " # # # # # #ˆ ‰

; therefore the equation is 1 or 1œ � œ � œ � œ625 25 75 x 2x16 16 25 75

y 16 y#

� �ˆ ‰ ˆ ‰ˆ ‰ ˆ ‰

3 34 4

2516

75

# ## #

#

8. The center is (0 0) and c 2 4 a b b 4 a . The equation is 1 1ß œ Ê œ � Ê œ � � œ Ê � œ# # # # ya b a b

x 49 144#

# # # #

#

1 49 4 a 144a a 4 a 196 49a 144a 4a a a 197a 196Ê � œ Ê � � œ � Ê � � œ � Ê � �49 144a 4 a# #a b�

# # # # # # # % % #a b a b 0 a 196 a 1 0 a 14 or a 1; a 14 b 4 (14) 0 which is impossible; a 1œ Ê � � œ Ê œ œ œ Ê œ � � œa b a b# # # #

b 4 1 3; therefore the equation is y 1Ê œ � œ � œ# # x3

#

9. b x a y a b ; at (x y ) the tangent line is y y (x x )# # # # # #" " " "� œ Ê œ � ß � œ � �dy

dx a y a yb x b x#

# #

#"

"

Š ‹ a yy b xx b x a y a b b xx a yy a b 0Ê � œ � œ Ê � � œ# # # # # # # # # # # #

" " " "" "

10. b x a y a b ; at (x y ) the tangent line is y y (x x )# # # # # #" " " "� œ Ê œ ß � œ �dy

dx a y a yb x b x#

# #

#"

"

Š ‹ b xx a yy b x a y a b b xx a yy a b 0Ê � œ � œ Ê � � œ# # # # # # # # # # # #

" " " "" "

11. 12.

13. 14.



15. 9x 4y 36 4x 9y 16 0a b a b# # # #� � � � Ÿ

9x 4y 36 0 and 4x 9y 16 0Ê � � Ÿ � � # # # #

or 9x 4y 36 0 and 4x 9y 16 0# # # #� � � � Ÿ

16. 9x 4y 36 4x 9y 16 0, which is thea b a b# # # #� � � � �

complement of the set in Exercise 15

17. (a) x e cos t and y e sin t x y e cos t e sin t e . Also tan tœ œ Ê � œ � œ œ œ2t 2t 4t 4t 4t# # # # yx e cos t

e sin t2t

2t

t tan x y e is the Cartesian equation. Since r x y andÊ œ Ê � œ œ ��" # # # # #ˆ ‰yx

% Îtan y x�" a b

tan , the polar equation is r e or r e for r 0) œ œ œ ��" #ˆ ‰yx

4 2) )

(b) ds r d dr ; r e dr 2e d# # # #œ � œ Ê œ) )2 2) )

ds r d 2e d e d 4e dÊ œ � œ �# # # # ## #) ) ) )ˆ ‰ ˆ ‰2 2 4) ) )

5e d ds 5 e d L 5 e dœ Ê œ Ê œ4 2 2) ) )) ) )# È È'0

21

e 1œ œ �’ “ a bÈ È5 e 52

42) #

! #

11

18. r 2 sin dr 2 sin cos d ds r d dr 2 sin d 2 sin cos dœ Ê œ Ê œ � œ �$ # # # # # $ # ## #ˆ ‰ ˆ ‰ ˆ ‰ � ‘ � ‘ˆ ‰ ˆ ‰ ˆ ‰) ) ) ) ) )3 3 3 3 3 3) ) ) )

4 sin d 4 sin cos d 4 sin sin cos d 4 sin dœ � œ � œ' # % # # % # # # % #ˆ ‰ ˆ ‰ ˆ ‰ � ‘ � ‘ ˆ ‰ˆ ‰ ˆ ‰ ˆ ‰) ) ) ) ) ) )3 3 3 3 3 3 3) ) ) )

ds 2 sin d . Then L 2 sin d 1 cos d sin 3Ê œ œ œ � œ � œ# # $

!ˆ ‰ ˆ ‰ � ‘ � ‘ˆ ‰ ˆ ‰) ) ) ) 1

3 3 3 2 32 3 2) ) ) ) 1' '

0 0

3 31 1

19. e 2 and r cos 2 x 2 is the directrix k 2; the conic is a hyperbola with rœ œ Ê œ Ê œ œ) ke1 e cos � )

rÊ œ œ(2)(2)1 2 cos 1 2 cos

4� �) )

20. e 1 and r cos 4 x 4 is the directrix k 4; the conic is a parabola with rœ œ � Ê œ � Ê œ œ) ke1 e cos � )

rÊ œ œ(4)(1)1 cos 1 cos

4� �) )

21. e and r sin 2 y 2 is the directrix k 2; the conic is an ellipse with rœ œ Ê œ Ê œ œ"# �) ke

1 e sin )

rÊ œ œ2

1 sin 2

2 sin

ˆ ‰ˆ ‰

"

#

"

#� �) )

22. e and r sin 6 y 6 is the directrix k 6; the conic is an ellipse with rœ œ � Ê œ � Ê œ œ"�3 1 e sin

ke))

rÊ œ œ6

1 sin 6

3 sin

ˆ ‰ˆ ‰

"

"

3

3� �) )



23. Arc PF Arc AF since each is the distance rolled;œ

PCF Arc PF b( PCF); n œ Ê œ n œArc PF Arc AFb a)

Arc AF a a b( PCF) PCF ;Ê œ Ê œ n Ê n œ) ) )ˆ ‰ab

OCB and OCB PCF PCEn œ � n œ n �n1# )

PCF œ n � � œ � � Ê �ˆ ‰ ˆ ‰ ˆ ‰1 1 1# # #! ) ! )a

b

œ � � Ê � œ � �ˆ ‰ ˆ ‰ ˆ ‰a ab b) ! ) ) !1 1 1

# # #

.Ê œ � � Ê œ �! 1 ) ) ! 1 )ˆ ‰ ˆ ‰a a bb b

�

Now x OB BD OB EP (a b) cos b cos (a b) cos b cosœ � œ � œ � � œ � � �) ! ) 1 )ˆ ‰ˆ ‰a bb�

(a b) cos b cos cos b sin sin (a b) cos b cos andœ � � � œ � �) 1 ) 1 ) ) )ˆ ‰ ˆ ‰ ˆ ‰ˆ ‰ ˆ ‰ ˆ ‰a b a b a bb b b� � �

y PD CB CE (a b) sin b sin (a b) sin b sinœ œ � œ � � œ � �) ! ) )ˆ ‰ˆ ‰a bb�

(a b) sin b sin cos b cos sin (a b) sin b sin ;œ � � � œ � �) 1 ) 1 ) ) )ˆ ‰ ˆ ‰ ˆ ‰ˆ ‰ ˆ ‰ ˆ ‰a b a b a bb b b� � �

therefore x (a b) cos b cos and y (a b) sin b sinœ � � œ � �) ) ) )ˆ ‰ ˆ ‰ˆ ‰ ˆ ‰a b a bb b� �

24. x a(t sin t) a(1 cos t) and let 1 dm dA y dx y dtœ � Ê œ � œ Ê œ œ œdx dxdt dt$ ˆ ‰

a(1 cos t) a (1 cos t) dt a (1 cos t) dt; then A a (1 cos t) dtœ � � œ � œ �# # # #'0

21

a 1 2 cos t cos t dt a 1 2 cos t cos 2t dt a t 2 sin tœ � � œ � � � œ � �# # # #" "# #

#

!' '

0 0

2 21 1a b ˆ ‰ � ‘3 sin 2t2 4

1

3 a ; x = x a(t sin t) and y = y a(1 cos t) M y dm y dAœ œ � œ � Ê œ œµ µ µ µ1 $# " "# # x ' '

a(1 cos t) a (1 cos t) dt a (1 cos t) dt 1 3 cos t 3 cos t cos t dtœ � � œ � œ � � �' ' '0 0 0

2 2 21 1 1

" "# # #

# # $ $ # $a$ a b 1 3 cos t 1 sin t (cos t) dt t 3 sin t sin tœ � � � � � œ � � � �a 3 3 cos 2t a 5 3 sin 2t sin t

2 4 3

$ $ $

# # # ##

#

!

'0

21� ‘a b ’ “ 1

. Therefore y a. Also, M x dm x dAœ œ œ œ œ œµ µ5 a 5MM 3 a 6 y

11

$

$

#

##x

5 aŠ ‹1

' ' $

a(t sin t) a (1 cos t) dt a t 2t cos t t cos t sin t 2 sin t cos t sin t cos t dtœ � � œ � � � � �' '0 0

2 21 1

# # $ # #a b a 2 cos t 2t sin t t cos 2t sin 2t cos t sin t 3 a . Thusœ � � � � � � � � œ$ # # # $" "

#

!’ “t t cos t

2 4 8 4 3

# $1

1

x a a a is the center of mass.œ œ œ Ê ßMM 3 a 6

3 a 5y 11

# $

# 1 1ˆ ‰

25. tan tan ( ) ;" < < " < <œ � Ê œ � œ# " # "�

�tan tan

1 tan tan < <

< <# "

# "

the curves will be orthogonal when tan is undefined, or"

when tan <#�" �"œ Ê œtan g ( )

r< )"

w

w’ “rf ( ))

r f ( ) g ( )Ê œ �# w w) )

26. r sin sin cos tan tanœ Ê œ Ê œ œ% $ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰) ) ) ))4 d 4 4 4

dr sin

sin cos<

%

$

ˆ ‰ˆ ‰ ˆ ‰

)

) )

4

4 4

27. r 2a sin 3 6a cos 3 tan tan 3 ; when , tan tan œ Ê œ Ê œ œ œ œ œ Ê œ) ) < ) ) < <dr r 2a sin 3d 6a cos 3 3 6 3) )

) 1 1 1ˆ ‰dr

d)

" "# #



28. (a) (b) r 1 r tan ) ) ) <œ Ê œ Ê œ � Ê�" �#drd) k

)œ1

lim tan œ œ � Ê œ �_))

�"

�#� ) <) Ä _

from the right as the spiral winds inÊ Ä< 1#

around the origin.

29. tan cot is at ; tan tan is 3 at ; since the product of< ) ) < ) )" #�

"œ œ � � œ œ œ œÈÈ È

3 cos 3 sin 3 3 cos 3

sin )

)

1 ) 1)

È these slopes is 1, the tangents are perpendicular�

30. tan is 1 at < ) <œ œ œ Ê œr a(1 cos )a sin 4ˆ ‰dr

d)

�#

)

)1 1



NOTES:


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